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Kinetic SIS opinion-driven models with asymmetric awareness feedback: macroscopic limit and polarization

Juan Pablo Pinasco, Nicolas Saintier, Horacio Tettamanti, Mattia Zanella

Abstract

We study a kinetic multi-agent framework coupling opinion dynamics with epidemic spreading, where individual social behaviour both affects and is affected by disease transmission. Each agent is characterised by an epidemiological state and a continuous opinion variable measuring compliance with non-pharmaceutical interventions. The key mechanism of the model is an asymmetric opinion update driven by epidemic encounters: infection events induce more cautious attitudes, while failed transmissions push individuals toward more extreme opinions. We focus on a prototypical SIS setting, for which we derive a macroscopic kinetic description and, in a fast social-interaction regime, a reduced system of differential equations capturing the feedback between epidemic prevalence and opinion evolution. Convergence of the reduced model is rigorously quantified through a modified Wasserstein distance. Numerical simulations highlight how infection-induced awareness and non-infection-driven extremization jointly shape collective epidemic-opinion dynamics.

Kinetic SIS opinion-driven models with asymmetric awareness feedback: macroscopic limit and polarization

Abstract

We study a kinetic multi-agent framework coupling opinion dynamics with epidemic spreading, where individual social behaviour both affects and is affected by disease transmission. Each agent is characterised by an epidemiological state and a continuous opinion variable measuring compliance with non-pharmaceutical interventions. The key mechanism of the model is an asymmetric opinion update driven by epidemic encounters: infection events induce more cautious attitudes, while failed transmissions push individuals toward more extreme opinions. We focus on a prototypical SIS setting, for which we derive a macroscopic kinetic description and, in a fast social-interaction regime, a reduced system of differential equations capturing the feedback between epidemic prevalence and opinion evolution. Convergence of the reduced model is rigorously quantified through a modified Wasserstein distance. Numerical simulations highlight how infection-induced awareness and non-infection-driven extremization jointly shape collective epidemic-opinion dynamics.
Paper Structure (20 sections, 5 theorems, 138 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 5 theorems, 138 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Kinetic approach to compartmental epidemiology
  3. Microscopic interaction dynamics
  4. The kinetic model
  5. Well-posedness
  6. Evolution of the moments.
  7. Derivation of reduced-complexity systems
  8. Zero-diffusion case: formal derivation of the macroscopic system
  9. Rigorous derivation of the macroscopic model
  10. Macroscopic SIS model in the presence of self-thinking dynamics
  11. Numerical Results
  12. Asymptotic behavior of the Kinetic Opinion Model
  13. Consistency with the evolution of macroscopic quantities
  14. Impact of reaction-type dynamics on epidemic evolution
  15. Conclusions
  16. ...and 5 more sections

Key Result

Theorem 2.1

For any initial condition $f_0^S,f_0^I\in \mathcal{M}_+([0,1])$ such that $f_0=f_0^S+f_0^I\in \mathcal{P}([0,1])$ there exists a unique solution $f^S, f^I\in C([0,+\infty),\mathcal{M}_+([0,1]))\cap C^1([0,+\infty),\mathcal{M}([0,1]))$ with $f_t=f_t^S+f_t^I\in \mathcal{P}([0,1])$. Here $\mathcal{M}([

Figures (6)

  • Figure 1: Equilibrium state \ref{['eq.state']} for different choices of the distribution parameter. When the self-thinking forces dominate the dynamics, we can observe a polarized distribution of opinions, while consensus formation emerges when the aggregation tendency surpasses the self-thinking effect.
  • Figure 2: Comparison between the DSMC solution of the Boltzmann-type model and the Fokker-Planck solution in the quasi-invariant regime. We consider different combinations of initial distributions and the self-thinking parameter. In particular we have in (a): $m(0)=0$, $\sigma^2=0.25$; in (b): $m(0)=0$, $\sigma^2=2$; in (c): $m(0)=0.3$, $\sigma^2=0.25$; in (d): $m(0)=0.3$, $\sigma^2=2$; in (e): $m(0)=0$, $\sigma^2=1$. We implemented the DSMC scheme with $N=10^{6}$ agents, a final time $T=5$ and $\epsilon = 10^{-1},10^{-3}$.
  • Figure 3: Implementation of the DSMC scheme in the limit where the self-thinking forces vanish. We consider as initial distribution a uniform distribution in the interval $[-1,1]$ and we take $\sigma^{2} = 10^{-1}$ (left) and $\sigma^{2} = 10^{-4}$ (right). We implemented the DSMC scheme with $N=10^{6}$ agents, a final time $T=5$ and $\epsilon = 10^{-3}$. In the limit $\sigma^{2} \to 0$ we obtain global consensus and all agents share the same opinion giving place to the formation of a singularity.
  • Figure 4: Evolution of macroscopic quantites obtained by the time evolution of (\ref{['Approx_Dirac']})-(\ref{['HighFreqGrazing2']}) and the DSMC scheme shown in Algorithm \ref{['algo:DSMC']}. Simulations in the first row are obtained using parameter set A (Table \ref{['tab:parameters']}), while those in the second row correspond to parameter set B. We consider $N=10^6$ agents between $t =[0,5]$ with $\Delta t=10^{-3}$ and different values of the rate of social interactions $\tau_{S} = 1,10^{-1},10^{-3}$.
  • Figure 5: Contagion probability in terms of the total population mean opinion for different values of the parameter $\alpha$. Its maximum value is equal to $\beta$ which in this case is set to $0.1$.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem C.1
  • proof
  • Theorem C.2
  • proof